Wiman--Valiron type inequalities for entire and random entire functions of finite logarithmic order (Q2773663)

Given an entire function \(f(z)=\sum_{n=0}^\infty a_n z^n\), put \(M_f(r)=\max\{| f(z)| :| z| =r\}\), \(\mu_f(r)=\max\{| a_n| r^n:n\geq 0\}\), and \(\nu_f(r)=\{n\geq 0:| a_n| r^n=\mu_f(r)\}\). The main results of the article are connected with the inequality NEWLINE\[NEWLINE \underset{r\to\infty}{\overline\lim} \frac{\ln M_f(r)-\ln \mu_f(r)}{\ln\ln\mu_f(r)}\leq \alpha. \eqno{(1)} NEWLINE\]NEWLINE The author establishes several necessary and sufficient conditions for the validity of inequality (1) in terms of the behavior of \(\mu_f(r)\) and \(\nu_f(r)\). The typical of them can be stated as follows. Let \(l\) be a real-valued positive logarithmically convex function such that \(\ln r=o(l(r))\) and let \(\alpha\in (0,\infty)\). Inequality (1) holds for every transcendental entire function \(f\) such that \(\ln\mu_f(r)\leq l(r)\) (\(r\geq r_0\)) if and only if \(\underset{r\to\infty}{\overline\lim}\frac{\ln l(r)}{\ln\ln r}\leq \alpha+1\). Similar statements are also proven for random entire functions. Next, the author studies the question of behavior of \(M_f(r)\) in dependence on the quantities \(\text{arg}\,a_n\).